Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Science China Information Sciences, Jun 2018

Although the mean square stabilization of hybrid systems by feedback control based on discretetime observations of state and mode has been studied by several authors since 2013, the corresponding almost sure stabilization problem has received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system ẋ(t) = f(x(t)) by a linear discrete-time stochastic feedback control Ax([t/τ]τ)dB/(t) (namely the stochastically controlled system has the form dx(t) = f(x(t))dt + Ax([t/τ]τ)dB/(t), where B(t) is a scalar Brownian, τ > 0, and [t/τ] is the integer part of t/τ. In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system ẋ(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ)dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f(x(t), r(t))dt + u(x([t/τ]τ))dB(t), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.

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Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Citation Song G F, Lu Z Y, Zheng B-C, et al. Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state. Sci China Inf Sci Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state Gongfei SONG 2 Zhenyu LU 1 Bo-Chao ZHENG 2 Xuerong MAO 0 0 Department of Mathematics and Statistics, University of Strathclyde , Glasgow G1 1XH , UK 1 School of Electronic and Information Engineering, Nanjing University of Information Science and Technology , Nanjing 210044 , China 2 Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology, School of Information and Control, Nanjing University of Information Science and Technology , Nanjing 210044 , China Although the mean square stabilization of hybrid systems by feedback control based on discretetime observations of state and mode has been studied by several authors since 2013, the corresponding almost sure stabilization problem has received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system x˙ (t) = f (x(t)) by a linear discrete-time stochastic feedback control Ax([t/τ ]τ )dB(t) (namely the stochastically controlled system has the form dx(t) = f (x(t))dt + Ax([t/τ ]τ )dB(t)), where B(t) is a scalar Brownian, τ > 0, and [t/τ ] is the integer part of t/τ . In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system x˙ (t) = f (x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ ]τ ), r([t/τ ]τ ))dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f (x(t), r(t))dt + u(x([t/τ ]τ ), r([t/τ ]τ ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain. Brownian motion; Markov chain; generalized Itoˆ formula; almost sure exponential stability; stochastic feedback control Introduction In recent years, stochastic systems have been considered by many researchers since many practical systems can be modeled using these kinds of systems. Many significant results for stochastic systems have been reported (see [1–13]). Markovian jump systems are a special class of hybrid stochastic systems, which can be found in some engineering systems including power systems, manufacturing systems, ecosystems, and so forth. The literature in this area is huge and lots of papers are open access, thus we only mention a few [14–18]. Shaikhet [19] provided the sufficient conditions of asymptotic mean square stability for Markovian systems with delay. Mao [20] discussed the problem of exponential stability of general nonlinear Markovian jump systems. As is well known, a given unstable system can be stabilized by noise or noise can be used to make a system more stable when it is already stable. Arnold et al. [21] pointed out that a linear system can be stabilized by zero mean stationary parameter noise. In [22], a linear hybrid stochastic system was stabilized by Gaussian type noise. In addition, Khasminskii [23] proposed that a system was stabilized by using two types of white noise. It was shown in [24] that an unstable nonlinear system can be stabilized by Brownian motion provided the growth condition is linear. Mao [25] showed that any nonlinear system x˙ (t) = f (x(t), t) whose coefficient satisfied the condition |f (x, t)| 6 K|x|, K > 0, it was possible to use the Brownian motions to stabilize the system. It is worth noting that Appleby et al. [26] presented a general theory on the problem of stochastic stabilization for a nonlinear functional differential equation by noise. Mao et al. [27] developed an unstable Markovian jump system x˙ (t) = f (x(t), r(t), t) that can be stabilized by stochastic control and the partial subsystem was controlled. In other words, the space S of the Markov chain was divided into two proper subspaces S1 and S2, i.e., S = S1 ∪ S2. In summary, Mao et al. [27] considered the controlled stochastic system dx(t) = f (x(t), r(t), t)dt + u(r(t), t)dB(t), ( 1 ) where u(i, t) = 0 for i ∈ S1 while u(i, t) = u(i, x(t)) was a feedback control for i ∈ S2. New methods and sufficient conditions on the stochastic stabilization for Markovian jump systems were provided in [28]. With some applications, two examples on stabilization and destabilization by noise in the plane were presented in [29]. We should of course point out that the corresponding problem based on discrete-time state observations has already been studied by some authors. Recently, Mao [30] was the first to study this stabilization problem. He also obtained a bound τ ∗ on τ for the controlled system to be stable as long as τ < τ ∗ (plus some other conditions of course). Here τ > 0 is the duration between two consecutive observations. From the point of control cost, it is clearly better to have a larger τ ∗. Influenced by [30], a number o (...truncated)


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Gongfei Song, Zhenyu Lu, Bo-Chao Zheng, Xuerong Mao. Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state, Science China Information Sciences, 2018, Volume 61, Issue 7, DOI: 10.1007/s11432-017-9297-1